Econometrics_Slide05.pdf

Introductory Econometrics ECON2206/ECON3209 S2, 2009 Slides05 Lecturer: Minxian Yang ie_Slides05 my, School of Economics, UNSW 1 5. Multiple Regression Model: Asymptotics (Ch5) 5. Multiple Regression Model: Asymptotics ? Lecture plan – Why large-sample properties (asymptotics) – Consistency of the OLS estimators – Asymptotic normality of the OLS estimators ie_Slides05 my, School of Economics, UNSW 2 5. Multiple Regression Model: Asymptotics (Ch5) ? What we need for inference – We need the sampling distribution of the OLS estimators a) MLR1-4 imply the OLS estimators are unbiased. b) MLR1-6 (CLM) imply the OLS estimators are normally distributed. c) The normality leads to the exact distributions of the t-stat and the F-stat, which are a basis for inference. – MLR6 (u ~ iid Normal) is often too strong an assumption in practice. ? Without MLR6, the results in b) and c) no longer holds. ? But they hold approximately for large samples. ? Inference will be based on large-sample approximation. ie_Slides05 my, School of Economics, UNSW 3 5. Multiple Regression Model: Asymptotics (Ch5) ? Asymptotic (large-sample) analysis – Reluctant to assume MLR6, we proceed as follows. ? find the asymptotic distribution of the estimators (the sampling distribution when n goes to infinity). ? use the asymptotic distribution to approximate the sampling distribution of the estimators. – The strategy will work if ? the asymptotic distribution is available, and ? the sample size n is large. – The strategy does work for the OLS estimators under MLR1-5. ie_Slides05 my, School of Economics, UNSW 4 5. Multiple Regression Model: Asymptotics (Ch5) ? Consistency – Let be an estimator for parameter βj, from a sample of size n. – is consistent for βj if and only if tends to zero as n goes infinity. (also Appendix C) – Consistency comes from the LLN (law of large numbers). ie_Slides05 my, School of Economics, UNSW 5 jβ? jβ? ) from differes ?( jjP ββ 5. Multiple Regression Model: Asym